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Theorem itg1addlem4 25208
Description: Lemma for itg1add 25211. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
Hypotheses
Ref Expression
i1fadd.1 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
i1fadd.2 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))
itg1add.4 𝑃 = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
Assertion
Ref Expression
itg1addlem4 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   πœ‘,𝑖,𝑗,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)
Proof of Theorem itg1addlem4
Dummy variables 𝑀 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
2 i1fadd.2 . . . . 5 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
31, 2i1fadd 25204 . . . 4 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
4 itg1add.4 . . . . . . 7 𝑃 = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
5 ax-addf 11186 . . . . . . . . 9 + :(β„‚ Γ— β„‚)βŸΆβ„‚
6 ffn 6715 . . . . . . . . 9 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ + Fn (β„‚ Γ— β„‚))
75, 6ax-mp 5 . . . . . . . 8 + Fn (β„‚ Γ— β„‚)
8 i1frn 25186 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
91, 8syl 17 . . . . . . . . 9 (πœ‘ β†’ ran 𝐹 ∈ Fin)
10 i1frn 25186 . . . . . . . . . 10 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
112, 10syl 17 . . . . . . . . 9 (πœ‘ β†’ ran 𝐺 ∈ Fin)
12 xpfi 9314 . . . . . . . . 9 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
139, 11, 12syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
14 resfnfinfin 9329 . . . . . . . 8 (( + Fn (β„‚ Γ— β„‚) ∧ (ran 𝐹 Γ— ran 𝐺) ∈ Fin) β†’ ( + β†Ύ (ran 𝐹 Γ— ran 𝐺)) ∈ Fin)
157, 13, 14sylancr 588 . . . . . . 7 (πœ‘ β†’ ( + β†Ύ (ran 𝐹 Γ— ran 𝐺)) ∈ Fin)
164, 15eqeltrid 2838 . . . . . 6 (πœ‘ β†’ 𝑃 ∈ Fin)
17 rnfi 9332 . . . . . 6 (𝑃 ∈ Fin β†’ ran 𝑃 ∈ Fin)
1816, 17syl 17 . . . . 5 (πœ‘ β†’ ran 𝑃 ∈ Fin)
19 difss 4131 . . . . 5 (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃
20 ssfi 9170 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃) β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
2118, 19, 20sylancl 587 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
22 ffun 6718 . . . . . . . . . . 11 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ Fun + )
235, 22ax-mp 5 . . . . . . . . . 10 Fun +
24 i1ff 25185 . . . . . . . . . . . . . . 15 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
251, 24syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
2625frnd 6723 . . . . . . . . . . . . 13 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
27 ax-resscn 11164 . . . . . . . . . . . . 13 ℝ βŠ† β„‚
2826, 27sstrdi 3994 . . . . . . . . . . . 12 (πœ‘ β†’ ran 𝐹 βŠ† β„‚)
29 i1ff 25185 . . . . . . . . . . . . . . 15 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
302, 29syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
3130frnd 6723 . . . . . . . . . . . . 13 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
3231, 27sstrdi 3994 . . . . . . . . . . . 12 (πœ‘ β†’ ran 𝐺 βŠ† β„‚)
33 xpss12 5691 . . . . . . . . . . . 12 ((ran 𝐹 βŠ† β„‚ ∧ ran 𝐺 βŠ† β„‚) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
3428, 32, 33syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
355fdmi 6727 . . . . . . . . . . 11 dom + = (β„‚ Γ— β„‚)
3634, 35sseqtrrdi 4033 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
37 funfvima2 7230 . . . . . . . . . 10 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
3823, 36, 37sylancr 588 . . . . . . . . 9 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
39 opelxpi 5713 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
4038, 39impel 507 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺)))
41 df-ov 7409 . . . . . . . 8 (π‘₯ + 𝑦) = ( + β€˜βŸ¨π‘₯, π‘¦βŸ©)
424rneqi 5935 . . . . . . . . 9 ran 𝑃 = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
43 df-ima 5689 . . . . . . . . 9 ( + β€œ (ran 𝐹 Γ— ran 𝐺)) = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
4442, 43eqtr4i 2764 . . . . . . . 8 ran 𝑃 = ( + β€œ (ran 𝐹 Γ— ran 𝐺))
4540, 41, 443eltr4g 2851 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ ran 𝑃)
4625ffnd 6716 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn ℝ)
47 dffn3 6728 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
4846, 47sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
4930ffnd 6716 . . . . . . . 8 (πœ‘ β†’ 𝐺 Fn ℝ)
50 dffn3 6728 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:β„βŸΆran 𝐺)
5149, 50sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐺:β„βŸΆran 𝐺)
52 reex 11198 . . . . . . . 8 ℝ ∈ V
5352a1i 11 . . . . . . 7 (πœ‘ β†’ ℝ ∈ V)
54 inidm 4218 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5545, 48, 51, 53, 53, 54off 7685 . . . . . 6 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆran 𝑃)
5655frnd 6723 . . . . 5 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran 𝑃)
5756ssdifd 4140 . . . 4 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}))
5826sselda 3982 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
5931sselda 3982 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6058, 59anim12dan 620 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
61 readdcl 11190 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ (𝑦 + 𝑧) ∈ ℝ)
6260, 61syl 17 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 + 𝑧) ∈ ℝ)
6362ralrimivva 3201 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
64 funimassov 7581 . . . . . . . 8 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ ↔ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6523, 36, 64sylancr 588 . . . . . . 7 (πœ‘ β†’ (( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ ↔ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6663, 65mpbird 257 . . . . . 6 (πœ‘ β†’ ( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ)
6744, 66eqsstrid 4030 . . . . 5 (πœ‘ β†’ ran 𝑃 βŠ† ℝ)
6867ssdifd 4140 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))
69 itg1val2 25193 . . . 4 (((𝐹 ∘f + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 βˆ– {0}) ∈ Fin ∧ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}) ∧ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))) β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
703, 21, 57, 68, 69syl13anc 1373 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
7130adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
7211adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
73 inss2 4229 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
7473a1i 11 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧}))
75 i1fima 25187 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 β†’ (◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol)
761, 75syl 17 . . . . . . . . . 10 (πœ‘ β†’ (◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol)
77 i1fima 25187 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
782, 77syl 17 . . . . . . . . . 10 (πœ‘ β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
79 inmbl 25051 . . . . . . . . . 10 (((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol ∧ (◑𝐺 β€œ {𝑧}) ∈ dom vol) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8076, 78, 79syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8180ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8219, 67sstrid 3993 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† ℝ)
8382sselda 3982 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ ℝ)
8483adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ ℝ)
8559adantlr 714 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
8684, 85resubcld 11639 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
8784recnd 11239 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ β„‚)
8885recnd 11239 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
8987, 88npcand 11572 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
90 eldifsni 4793 . . . . . . . . . . . . 13 (𝑀 ∈ (ran 𝑃 βˆ– {0}) β†’ 𝑀 β‰  0)
9190ad2antlr 726 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 β‰  0)
9289, 91eqnetrd 3009 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0)
93 oveq12 7415 . . . . . . . . . . . . 13 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = (0 + 0))
94 00id 11386 . . . . . . . . . . . . 13 (0 + 0) = 0
9593, 94eqtrdi 2789 . . . . . . . . . . . 12 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 0)
9695necon3ai 2966 . . . . . . . . . . 11 (((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0 β†’ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0))
9792, 96syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0))
98 itg1add.3 . . . . . . . . . . 11 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))
991, 2, 98itg1addlem3 25207 . . . . . . . . . 10 ((((𝑀 βˆ’ 𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0)) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
10086, 85, 97, 99syl21anc 837 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
1011, 2, 98itg1addlem2 25206 . . . . . . . . . . 11 (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
102101ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
103102, 86, 85fovcdmd 7576 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
104100, 103eqeltrrd 2835 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
10571, 72, 74, 81, 104itg1addlem1 25201 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
10683recnd 11239 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ β„‚)
1071, 2i1faddlem 25202 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ β„‚) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
108106, 107syldan 592 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
109108fveq2d 6893 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀})) = (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
110100sumeq2dv 15646 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
111105, 109, 1103eqtr4d 2783 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀})) = Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧))
112111oveq2d 7422 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = (𝑀 Β· Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧)))
113103recnd 11239 . . . . . 6 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
11472, 106, 113fsummulc2 15727 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
115112, 114eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
116115sumeq2dv 15646 . . 3 (πœ‘ β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
11787, 113mulcld 11231 . . . . 5 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
118117anasss 468 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (ran 𝑃 βˆ– {0}) ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
11921, 11, 118fsumcom 15718 . . 3 (πœ‘ β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12070, 116, 1193eqtrd 2777 . 2 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
121 oveq1 7413 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦 + 𝑧) = ((𝑀 βˆ’ 𝑧) + 𝑧))
122 oveq1 7413 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦𝐼𝑧) = ((𝑀 βˆ’ 𝑧)𝐼𝑧))
123121, 122oveq12d 7424 . . . . . 6 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12418adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 ∈ Fin)
12567adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 βŠ† ℝ)
126125sselda 3982 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ ℝ)
12759adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑧 ∈ ℝ)
128126, 127resubcld 11639 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ)
129128ex 414 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ))
130126recnd 11239 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ β„‚)
131130adantrr 716 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑣 ∈ β„‚)
13267sselda 3982 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ ran 𝑃) β†’ 𝑦 ∈ ℝ)
133132ad2ant2rl 748 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ ℝ)
134133recnd 11239 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ β„‚)
13559recnd 11239 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
136135adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑧 ∈ β„‚)
137131, 134, 136subcan2ad 11613 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦))
138137ex 414 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦)))
139129, 138dom2lem 8985 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ)
140 f1f1orn 6842 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
141139, 140syl 17 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
142 oveq1 7413 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 βˆ’ 𝑧) = (𝑀 βˆ’ 𝑧))
143 eqid 2733 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))
144 ovex 7439 . . . . . . . 8 (𝑀 βˆ’ 𝑧) ∈ V
145142, 143, 144fvmpt 6996 . . . . . . 7 (𝑀 ∈ ran 𝑃 β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
146145adantl 483 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
147 f1f 6785 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„)
148 frn 6722 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„ β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βŠ† ℝ)
149139, 147, 1483syl 18 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βŠ† ℝ)
150149sselda 3982 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑦 ∈ ℝ)
15159adantr 482 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑧 ∈ ℝ)
152150, 151readdcld 11240 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦 + 𝑧) ∈ ℝ)
153101ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
154153, 150, 151fovcdmd 7576 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦𝐼𝑧) ∈ ℝ)
155152, 154remulcld 11241 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
156155recnd 11239 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
157123, 124, 141, 146, 156fsumf1o 15666 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
158125sselda 3982 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ ℝ)
159158recnd 11239 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ β„‚)
160135adantr 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑧 ∈ β„‚)
161159, 160npcand 11572 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
162161oveq1d 7421 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
163162sumeq2dv 15646 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑀 ∈ ran 𝑃(((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
164157, 163eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
16536ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
166 simpr 486 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ran 𝐹)
167 simplr 768 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ran 𝐺)
168166, 167opelxpd 5714 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
169 funfvima2 7230 . . . . . . . . . . . 12 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
17023, 169mpan 689 . . . . . . . . . . 11 ((ran 𝐹 Γ— ran 𝐺) βŠ† dom + β†’ (βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
171165, 168, 170sylc 65 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺)))
172 df-ov 7409 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + β€˜βŸ¨π‘¦, π‘§βŸ©)
173171, 172, 443eltr4g 2851 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ran 𝑃)
17458adantlr 714 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
175174recnd 11239 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ β„‚)
176135adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
177175, 176pncand 11569 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) βˆ’ 𝑧) = 𝑦)
178177eqcomd 2739 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧))
179 oveq1 7413 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) β†’ (𝑣 βˆ’ 𝑧) = ((𝑦 + 𝑧) βˆ’ 𝑧))
180179rspceeqv 3633 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃 ∧ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧)) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
181173, 178, 180syl2anc 585 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
182181ralrimiva 3147 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
183 ssabral 4059 . . . . . . 7 (ran 𝐹 βŠ† {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)} ↔ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
184182, 183sylibr 233 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)})
185143rnmpt 5953 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)}
186184, 185sseqtrrdi 4033 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
18759adantr 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
188174, 187readdcld 11240 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ℝ)
189101ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
190189, 174, 187fovcdmd 7576 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
191188, 190remulcld 11241 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
192191recnd 11239 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
193149ssdifd 4140 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹) βŠ† (ℝ βˆ– ran 𝐹))
194193sselda 3982 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹)) β†’ 𝑦 ∈ (ℝ βˆ– ran 𝐹))
195 eldifi 4126 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ 𝑦 ∈ ℝ)
196195ad2antrl 727 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑦 ∈ ℝ)
19759adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑧 ∈ ℝ)
198 simprr 772 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
1991, 2, 98itg1addlem3 25207 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0)) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
200196, 197, 198, 199syl21anc 837 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
201 inss1 4228 . . . . . . . . . . . . . . 15 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦})
202 eldifn 4127 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ Β¬ 𝑦 ∈ ran 𝐹)
203202ad2antrl 727 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ 𝑦 ∈ ran 𝐹)
204 vex 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
205204eliniseg 6091 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦))
206205elv 3481 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦)
207 vex 3479 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
208204, 207brelrn 5940 . . . . . . . . . . . . . . . . . . 19 (𝑣𝐹𝑦 β†’ 𝑦 ∈ ran 𝐹)
209206, 208sylbi 216 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) β†’ 𝑦 ∈ ran 𝐹)
210203, 209nsyl 140 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ 𝑣 ∈ (◑𝐹 β€œ {𝑦}))
211210pm2.21d 121 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) β†’ 𝑣 ∈ βˆ…))
212211ssrdv 3988 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (◑𝐹 β€œ {𝑦}) βŠ† βˆ…)
213201, 212sstrid 3993 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† βˆ…)
214 ss0 4398 . . . . . . . . . . . . . 14 (((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† βˆ… β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = βˆ…)
215213, 214syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = βˆ…)
216215fveq2d 6893 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = (volβ€˜βˆ…))
217 0mbl 25048 . . . . . . . . . . . . . 14 βˆ… ∈ dom vol
218 mblvol 25039 . . . . . . . . . . . . . 14 (βˆ… ∈ dom vol β†’ (volβ€˜βˆ…) = (vol*β€˜βˆ…))
219217, 218ax-mp 5 . . . . . . . . . . . . 13 (volβ€˜βˆ…) = (vol*β€˜βˆ…)
220 ovol0 25002 . . . . . . . . . . . . 13 (vol*β€˜βˆ…) = 0
221219, 220eqtri 2761 . . . . . . . . . . . 12 (volβ€˜βˆ…) = 0
222216, 221eqtrdi 2789 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = 0)
223200, 222eqtrd 2773 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = 0)
224223oveq2d 7422 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) Β· 0))
225196, 197readdcld 11240 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ ℝ)
226225recnd 11239 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ β„‚)
227226mul01d 11410 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· 0) = 0)
228224, 227eqtrd 2773 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
229228expr 458 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ βˆ– ran 𝐹)) β†’ (Β¬ (𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0))
230 oveq12 7415 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = (0 + 0))
231230, 94eqtrdi 2789 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = 0)
232 oveq12 7415 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = (0𝐼0))
233 0re 11213 . . . . . . . . . . 11 0 ∈ ℝ
234 iftrue 4534 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) β†’ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))) = 0)
235 c0ex 11205 . . . . . . . . . . . 12 0 ∈ V
236234, 98, 235ovmpoa 7560 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) β†’ (0𝐼0) = 0)
237233, 233, 236mp2an 691 . . . . . . . . . 10 (0𝐼0) = 0
238232, 237eqtrdi 2789 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = 0)
239231, 238oveq12d 7424 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (0 Β· 0))
240 0cn 11203 . . . . . . . . 9 0 ∈ β„‚
241240mul01i 11401 . . . . . . . 8 (0 Β· 0) = 0
242239, 241eqtrdi 2789 . . . . . . 7 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
243229, 242pm2.61d2 181 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ βˆ– ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
244194, 243syldan 592 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
245 f1ofo 6838 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
246141, 245syl 17 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
247 fofi 9335 . . . . . 6 ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
248124, 246, 247syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
249186, 192, 244, 248fsumss 15668 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
25019a1i 11 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃)
251117an32s 651 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
252 dfin4 4267 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))
253 inss2 4229 . . . . . . . 8 (ran 𝑃 ∩ {0}) βŠ† {0}
254252, 253eqsstrri 4017 . . . . . . 7 (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) βŠ† {0}
255254sseli 3978 . . . . . 6 (𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ {0})
256 elsni 4645 . . . . . . . . 9 (𝑀 ∈ {0} β†’ 𝑀 = 0)
257256adantl 483 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 = 0)
258257oveq1d 7421 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
259101ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
260257, 233eqeltrdi 2842 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 ∈ ℝ)
26159adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑧 ∈ ℝ)
262260, 261resubcld 11639 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
263259, 262, 261fovcdmd 7576 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
264263recnd 11239 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
265264mul02d 11409 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
266258, 265eqtrd 2773 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
267255, 266sylan2 594 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
268250, 251, 267, 124fsumss 15668 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
269164, 249, 2683eqtr4d 2783 . . 3 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
270269sumeq2dv 15646 . 2 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
271192anasss 468 . . 3 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
27211, 9, 271fsumcom 15718 . 2 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
273120, 270, 2723eqtr2d 2779 1 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  {csn 4628  βŸ¨cop 4634  βˆͺ ciun 4997   class class class wbr 5148   ↦ cmpt 5231   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6535   Fn wfn 6536  βŸΆwf 6537  β€“1-1β†’wf1 6538  β€“ontoβ†’wfo 6539  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408   ∘f cof 7665  Fincfn 8936  β„‚cc 11105  β„cr 11106  0cc0 11107   + caddc 11110   Β· cmul 11112   βˆ’ cmin 11441  Ξ£csu 15629  vol*covol 24971  volcvol 24972  βˆ«1citg1 25124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185  ax-addf 11186
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xadd 13090  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-xmet 20930  df-met 20931  df-ovol 24973  df-vol 24974  df-mbf 25128  df-itg1 25129
This theorem is referenced by:  itg1addlem5  25210
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